Solutions and Stabilities for a 2D-Non Homogeneous Lane-Emden Fractional System

2018 ◽  
Vol 11 (2) ◽  
pp. 1-14 ◽  
Author(s):  
Zakaria Bekkouche ◽  
Zoubir Dahmani ◽  
Guo Zhang
Keyword(s):  
Fractional System

scholarly journals Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

2021 ◽  
Vol 11 (15) ◽  
pp. 6955
Author(s):  
Andrzej Rysak ◽  
Magdalena Gregorczyk
Keyword(s):  
Optimal Parameter ◽  
Differential Transform Method ◽  
Bifurcation Diagrams ◽  
Transform Method ◽  
Rossler System ◽  
Differential Transform ◽  
Rössler System ◽  
Fractional System ◽  
Parameter Values

This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.

scholarly journals A Fractional Approach to a Computational Eco-Epidemiological Model with Holling Type-II Functional Response

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1159
Author(s):  
B. Günay ◽  
Praveen Agarwal ◽  
Juan L. G. Guirao ◽  
Shaher Momani
Keyword(s):  
Numerical Method ◽  
Decision Makers ◽  
Derivative Operator ◽  
Research Fields ◽  
Fractional System ◽  
Chaotic Nature ◽  
And Control ◽  
The Impact ◽  
Distinguishing Features ◽  
Type Ii Functional Response

Eco-epidemiological can be considered as a significant combination of two research fields of computational biology and epidemiology. These problems mainly take ecological systems into account of the impact of epidemiological factors. In this paper, we examine the chaotic nature of a computational system related to the spread of disease into a specific environment involving a novel differential operator called the Atangana–Baleanu fractional derivative. To approximate the solutions of this fractional system, an efficient numerical method is adopted. The numerical method is an implicit approximate method that can provide very suitable numerical approximations for fractional problems due to symmetry. Symmetry is one of the distinguishing features of this technique compared to other methods in the literature. Through considering different choices of parameters in the model, several meaningful numerical simulations are presented. It is clear that hiring a new derivative operator greatly increases the flexibility of the model in describing the different scenarios in the model. The results of this paper can be very useful help for decision-makers to describe the situation related to the problem, in a more efficient way, and control the epidemic.

scholarly journals A Comparative Analysis of Fractional-Order Gas Dynamics Equations via Analytical Techniques

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1735
Author(s):  
Shuang-Shuang Zhou ◽  
Nehad Ali Shah ◽  
Ioannis Dassios ◽  
S. Saleem ◽  
Kamsing Nonlaopon
Keyword(s):  
Homotopy Perturbation Method ◽  
Graphical Representation ◽  
Analytical Techniques ◽  
Variational Iteration Method ◽  
Computational Techniques ◽  
System Of Nonlinear Equations ◽  
Gas Dynamics Equations ◽  
Homotopy Perturbation ◽  
Fractional System ◽  
Modified Forms

This article introduces two well-known computational techniques for solving the time-fractional system of nonlinear equations of unsteady flow of a polytropic gas. The methods suggested are the modified forms of the variational iteration method and the homotopy perturbation method by the Elzaki transformation. Furthermore, an illustrative scheme is introduced to verify the accuracy of the available techniques. A graphical representation of the exact and derived results is presented to show the reliability of the suggested approaches. It is also shown that the findings of the current methodology are in close harmony with the exact solutions. The comparative solution analysis via graphs also represents the higher reliability and accuracy of the current techniques.

Solitons propagation dynamics in a saturable PT-symmetric fractional medium

2021 ◽  
Author(s):  
Davood Hajitaghi Tehrani ◽  
Mehdi Solaimani ◽  
Mahboubeh Ghalandari ◽  
Bahman Babayar Razlighi
Keyword(s):  
Monte Carlo ◽  
Numerical Methods ◽  
Physical Properties ◽  
Maximum Intensity ◽  
Step Method ◽  
Saturation Parameter ◽  
Potential Depth ◽  
Fractional System ◽  
Nonlinear Fractional Schrödinger Equation ◽  
Good Agreement

Abstract In the current research, the propagation of solitons in a saturable PT-symmetric fractional system is studied by solving nonlinear fractional Schrödinger equation. Three numerical methods are employed for this purpose, namely Monte Carlo based Euler-Lagrange variational schema, split-step method, and extrapolation approach. The results show good agreement and accuracy. The effect of different parameters such as potential depth, Levy indices, and saturation parameter, on the physical properties of the systems such as maximum intensity and soliton width oscillations are considered.

Stochastic resonance for a fractional oscillator with random trichotomous mass and random trichotomous frequency

2017 ◽  
Vol 31 (30) ◽  
pp. 1750231 ◽  
Author(s):  
Lifeng Lin ◽  
Huiqi Wang ◽  
Suchuan Zhong
Keyword(s):  
Stochastic Resonance ◽  
Exact Expression ◽  
Order Moment ◽  
Periodic Signal ◽  
Transformation Technique ◽  
First Order ◽  
Fractional Oscillator ◽  
Trichotomous Noise ◽  
Output Amplitude ◽  
Fractional System

The stochastic resonance (SR) phenomena of a linear fractional oscillator with random trichotomous mass and random trichotomous frequency are investigate in this paper. By using the Shapiro–Loginov formula and the Laplace transformation technique, the exact expression of the first-order moment of the system’s steady response is derived. The numerical results demonstrate that the evolution of the output amplitude is nonmonotonic with frequency of the periodic signal, noise parameters and fractional order. The generalized SR (GSR) phenomena, including single GSR (SGSR) and doubly GSR (DGSR), and trebly GSR (TGSR), are detected in this fractional system. Then, the GSR regions in the [Formula: see text] plane are determined through numerical calculations. In addition, the interaction effect of the multiplicative trichotomous noise and memory can diversify the stochastic multiresonance (SMR) phenomena, and induce reverse-resonance phenomena.

Stability and resonance conditions of second-order fractional systems

2016 ◽  
Vol 24 (4) ◽  
pp. 659-672 ◽  
Author(s):  
Elena Ivanova ◽  
Xavier Moreau ◽  
Rachid Malti
Keyword(s):  
Second Order ◽  
Damping Factor ◽  
Fractional Differentiation ◽  
Resonance Amplitude ◽  
Fractional Systems ◽  
Integral Number ◽  
At Resonance ◽  
Fractional System ◽  
The Stability ◽  
Normalized Gain

The interest of studying fractional systems of second order in electrical and mechanical engineering is first illustrated in this paper. Then, the stability and resonance conditions are established for such systems in terms of a pseudo-damping factor and a fractional differentiation order. It is shown that a second-order fractional system might have a resonance amplitude either greater or less than one. Moreover, three abaci are given allowing the pseudo-damping factor and the differentiation order to be determined for, respectively, a desired normalized gain at resonance, a desired phase at resonance, and a desired normalized resonant frequency. Furthermore, it is shown numerically that the system root locus presents a discontinuity when the fractional differentiation order is an integral number.

scholarly journals Variational theory for (2+1)-dimensional fractional dispersive long wave equations

2021 ◽  
pp. 23-23
Author(s):  
Xiao-Qun Cao ◽  
Cheng-Zhuo Zhang ◽  
Shi-Cheng Hou ◽  
Ya-Nan Guo ◽  
Ke-Cheng Peng
Keyword(s):  
Porous Media ◽  
Nonlinear Waves ◽  
Inverse Method ◽  
Wave Equations ◽  
Continuous Medium ◽  
Variational Principles ◽  
Variational Theory ◽  
Long Wave ◽  
Potential Applications ◽  
Fractional System

This paper extends the (2+1)-dimensional Eckhaus-type dispersive long wave equations in continuous medium to their fractional partner, which is a model of nonlinear waves in fractal porous media. The derivation is shown briefly using He?s fractional derivative. Using the semi-inverse method, the variational principles are established for the fractional system, which up to now are not discovered. The obtained fractal variational principles are proved correct by minimizing the functionals with the calculus of variations, and might find potential applications in numerical modelling.

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